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Lightning surge propagation in overhead lines and gas insulated bus-ducts and cables Lee, Kai-Chung 1980-12-31

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LIGHTNING SURGE PROPAGATION IN OVERHEAD LINES AND GAS INSULATED BUS-DUCTS AND CABLES by |^LEE KAI-CHUNG B.ScT, University of Wisconsin, 1973 M.Sc., University of British Columbia, 1975 M.A.Sc., University of British Columbia, 1977 r A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY The Faculty of Graduate Studies in the Department '_ of Electrical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1980 0 Lee Kai-Chung, 1980 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of ____________________ The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1WS Date ABSTRACT The propagation characteristics of lightning surges in compressed SFg gas insulated power substation was studied using an electromagnetic transients program. Numerical models were developed to represent the behaviour of different system components especially under lightning over-voltage conditions. The characteristics of lightning surge propagation in overhead multi-phase untransposed transmission lines was analysed first. Modal analysis, :tpgether with special rotation techniques to fit time domain solutions were then used to simulate the wave propagation in multi-phase untransposed line in an electromagnetic transients program. Non-linear voltage-dependent corona attenuation and distortion phenomena were also investigated. Available field test results could be duplicated to within 5%. The characteristics of lightning surge propagation in multi-phase single-core SF^ cables was studied next. A program was developed to obtain the cable parameters for typical cable configurations. The amount of core current returning through its own sheath and through the earth were computed to illustrate the single phase cable representation for wave propagation in single core SFft cables. ii TABLE OF CONTENTS Page ABSTRACT ii TABLE OF CONTENTS iiACKNOWLEDGEMENTS vINTRODUCTION 1 CHAPTER 1 - LIGHTNING CHARACTERISTICS AND STROKES TO POWER TRANSMISSION LINES 1. Introduction 3 2. Lightning discharge mechanism 3 3. Statistical characteristics of lightning strokes 7 4. Frequency of lightning strokes to earth 9 5. Frequency of lightning strokes to power lines . . 12 6. Shielding failure phenomenon of lightning strokes 13 CHAPTER 2 - LIGHTNING SURGE PROPAGATION IN OVERHEAD TRANSMISSION LINES 1. Introduction 18 2. Modal analysis for N-phase untransposed line. . . 19 3. Rotation of eigenvectors for zero shunt conductance 22 4. Confirmation of accuracy of eigenvalue and eigenvector subroutine 25 5. Real-valued frequency-independent transformation matrix 26. Frequency dependent effect in lightning surge propagation 7 7. Determination of surge impedance of the struck phase of a transmission line 33 8. Single phase representation for close-by strokes on double circuited line 36 iii CHAPTER 3 - LIGHTNING WAVE PROPAGATION IN SF& GAS INSULATED UNDERGROUND TRANSMISSION CABLE SYSTEM 1. Introduction 46 2. Formation of series impedance matrix for SF, cables 9 6 3. Calculation of self and mutual earth return. ... 50 4. Calculation of self impedance matrix for single core cable. 61 5. Sheath current return characteristics for usual earth 67 6. Sheath current return characteristics for substation earth with grounding grid network ... 74 7. Formation of shunt admittance matrix for SFg cables 79 8. Confirmation of numerical accuracy for cable parameter calculation and current return ratio . . 80 9. Single phase representation parameters for multi-phase SF^ cables 80 10. Wave propagation in SF^ cables 84 CHAPTER 4 - CORONA ATTENUATION AND DISTORTION CHARACTERISTICS OF LIGHTNING OVERVOLTAGE IN OVERHEAD TRANSMISSION LINES 1. Introduction 87 2. Physical properties of corona attenuation and distortion characteristics 87 3. Transmission line equations for coronated lines. . 90 4. Solution of line equations by compensation method with trapezoidal rules 91 5. Influence on corona by adjacent sub-conductors in the same bundle 97 6. Influence on corona by adjacent phase conductors . 99 7. Optimal lumping locations and number of corona branch legs 99 8. Overall numerical modelling for corona effects . . 102 iv CHAPTER 5 - CONCLUSIONS 106 APPENDIX A - SKIN DEPTH ATTENUATION IN CONDUCTING MEDIUM WITH FINITE CONDUCTIVITY 107 BIBLIOGRAPHY 109 v ACKNOWLEDGEMENT I would like to show my deepest appreciation to my thesis supervisor Professor Hermann W. Dommel for providing such a unique chance in doing research in electromagnetic transients of power systems. Dr. Dommel's valuable criticism, and countless hours of discussion during research are also gratefully acknowledged. I am grateful to the B.C. Hydro Engineers, Messrs. Jack Sawada, Brent Hughes, Ken Nishikawara and Nick Cuk for their helpful discussions. I am also thankful to my fellow graduate students and colleagues Messrs. Obed Abledu, But-Chung Chiu, Shi Wei for their different points of view. The financial support from the University of British Columbia in form of teaching and research assistantship and fellowship is very much appreciated. The financial assistance of the Systems Engineering Division of the British Columbia Hydro and Power Authority through a Power Systems Research Agreement, and the British Columbia Telephone Company Graduate Scholarship are also gratefully acknowledged. Special thanks are expressed to Miss Gail Hrehorka in the Electrical Engineering Main Office for producing this excellently typed thesis. Finally, I am indebted to my parents for their continuous encouragement and my wife for her patience. vi 1 INTRODUCTION Every year, atmospheric lightning discharges cause numerous disturbances and damages to electric power systems, such as, destroying transformers and causing black-outs of large areas. This thesis is devoted to the analysis of lightning surge propagation into compressed SFg gas-insulated substations. The MICA project of the British Columbia Hydro and Power Authority was chosen as a test example. Insulation co-ordination requirements are usually derived from simulated surge propagation studies. This thesis shows that the present practice of insulation co-ordination design can be improved with the numerical models developed in this thesis. The contributions of this thesis to insulation co-ordination design and related power system studies includes the following: 1. Determination of wave propagation in untransposed lines - Analysis is used, with a special rotation of modal parameters and transformation matrices to make the method suitable for time-domain solutions of wave propagation in multi-phase untransposed line. The suitability of different simplified transmission line models is clarified by comparing simulation results with those from an exact multi-phase rep re sentat ion. 2. Representation of non-linear voltage-dependent corona effects - Corona distortion and attenuation has been simulated with voltage dependent 44 12 velocities and correction factors in the past ' , or with finite difference methods. However, these methods are inefficient for digital computer applications. More efficient computational algorithms, using compensation methods, are developed in this thesis to 2 investigate the non-linear voltage dependent corona effects. 3. Determination of wave propagation in multi-phase single core SF^ cables - Published methods for the calculation of cable constants give inconsistent results. A new cable constants program for multi phase single core SF^-cables has been developed by the author, using various converging.Mnfinite series. The complete shielding effect of the externally grounded sheath at frequencies above 1 k Hz has been confirmed with this program. The problem of transient groundrise caused by internal breakdowns or by lightning impulses, as studied by Ontario Hydro"^, is not included in this thesis. These transient potential differences between SF^ bus-ducts and ground occur mainly at the junction with the overhead line. As this thesis shows, the current return in the SF bus-duct is completely o through the sheath at frequencies above 1 kHz, whereas the current return of the lightning impulse on the overhead line isin the ground. At the junction, the return current must therefore pass from the sheath into the ground through the ground leads, which in turn causes the transient groundrise problem. These transient groundrises are an important factor in the design of the grounding system, because they can cause damage to ru auxiliary wiring or shocks to personnel. 3 CHAPTER 1: LIGHTNING CHARACTERISTICS AND STROKES TO POWER TRANSMISSION LINES 1. Introduction The first important experiment on lightning was done by Benjamin Franklin, who used flying kites to show that lightning is electrical in nature. For more than two centuries, lightning has been the subject of active research. Much of this research has been concerned with the pro-tection of people and property against the effects of lightning stroke. 2. Lightning discharge mechanisms Lightning strokes are first initiated inside thunder-clouds. A thunder-cloud usually contains several negative and positive charge centres distributed in different locations as shown in Figure 1.1a. As soon as the electron is jumping over to neutralize the positive charge, a step leader starts to move down the earth in discrete zig-zag steps of about 50 meters in length as shown in Figure 1.1b. This downward pilot stroke is about 1 cm in diameter and is not visually detectable by the human eye. As this stepped leader continues to progress downwards, positive charges are induced and accumulated on the ground surface. Eventually, these positive charges jump upwards and form the return stroke to meet the downward stepped leader as shown in Figure 1.1c. This highly luminous return stroke produces most of the thunder which is heard. The return stroke is about 10 cm in diameter and at a temperature of around 30,000°K. Once all the positive charges transfer to the thunder-cloud as shown in Figure l.ld, the discharged charge centre completely becomes 4 a. Charge neutralization b. Stepped leader moving within the cloud. downwards. Initialization of upward d. Complete upward propagation moving return leader. of return leader to cloud (charge centre becomes positive). Figure 1.1: Charge distribution and propagation during initial lightning discharge. 5 a. Discharge between 2 b. Negative charge dart charge centres. stroke flowing down the continuous earth path. c. Negative charge dart d. Formation of subsequent stroke about to hit return leader from ground the ground. o cloud charge centres. Figure 1.2: Charge distribution and propagation during sub sequent dart leader '(multipli-stroke lightning). 6 1 positive and single stroke lightning discharge is completed. 2 3 However, about 50% of all lightning flashes are multi-strokes ' and contain 3 or 4 subsequent strokes, typically separated by 30 to 40 ms. About less than 100 ms after the first stroke, a high potential difference is again established between the charge centres. Discharges again occur and a dart leader is formed which moves earthwards in the previous main stream as shown in Figure 1.2a to 1.2c. Similarly, a return stroke is also formed and more positive charges transfer to the thunder-clouds as shown in Figure 1.2d. The whole process of multi-strokes with relative stroke magnitudes and time scales is illustrated in Figure 1.3a and- 1.3b. Typical sub sequent strokes are of the initial stroke magnitude and are well separated (about 30 - 40 ms) in time. The initial stroke is the prime factor in the insulation co-ordination studies, but subsequent strokes must be taken into account as arresters must be able to handle repetitive discharges, and the dead times of the auto-reclosing switchgear must be set longer. Velocity 100% = 300 m/jjs Figure 1.3a: Diagram showing time intervals between initial and subsquent strokes.(Ref.3) 100|is lOQuis -35ms- •35ms Figure 1.3b: Current magnitudes of initial and subsequent strokes in typical lightning flashes. 3. Statistical characteristics of lightning strokes Due to the different distributions and intensities of charge centres inside the thunder-cloud, the characteristics of lightning strokes show a large statistical variation in both magnitude and shape. a. Magnitude of lightning strokes The voltage stress on the power system depends on the magnitude of the lightning current, which is therefore a critical 4 52 factor in determining insulation requirements. -Recorded measurements .' are shown in Figure 1.4. It can be seen that 80% of the lightning current magnitudes are within 10 to 100 kA, and only 5% exceed magnitude of 100 kA. 52 It is suggested that the lightning stroke has to be simulated as an incident current source to the power line with a maximum current magnitude of 100 kA for insulation co-ordination studies. However, this current source will become an overvoltage wave when propagating down the power line due to the inherent surge impedance of the line. Thus, for 8 Figure 1.4: Cumulative Probability of Occurrence of the Amplitudes of Lightning Currents obtained by summarising results from more than 624 measured incidents from 9 countries (Ref. 4). 9 equipment test purposes, overvoltage waves are prescribed, b. Waveshape of lightning stroke The lightning waveshapes measured by different researchers essen tially resemble a double exponential waveshape of different rise time and decay time. The observed spread of rise time is from very short to 10 us. 5 The observed decay time also spreads from 2 to 100 ps (see Figure 1.5a). The electric power industry therefore agreed many years ago to use a lightning overvoltage wave for equipment insulation testing purpose of a shape 1.2 x 50 ys (explanation-of designation in Figure 1.5b and 1.5c). Some testing precriptions also specify that this full wave be chopped with a spark gap in the tail to expose the equipment to the higher frequencies which are contained in the voltage collapse. 4. Frequency of lightning strokes to earth The thunderstorm activity on earth is measured by the isokeraunic level. This isokeraunic level (IKL) gives the number of days per year that thunder has been heard. Usually, thunder cannot be heard outside a 7-24 km radius. An updated world map of isokeraunic level is shown in Figure 1.6. As expected, higher IKL is found within the tropical and sub-tropical regions close to the equator. After obtaining the IKL of a given place, the number of strokes 2 7 to earth per km (N) in a particular location is given by N = A (IKL) stroke/(km2 - yr) where A = 0.1 to 0.2 10 t^=rise time =time to crest t2=decay time =time to half value 10 100 Time^s) a. Wave fronts and tails of lightning surges Time Double exponential wave V = v^ ( e -*t e r ) s wave Time to crest t1 =1.67(X2-X ) . =1.2jus Time to half value t =X.-X I 4 o =50JJS Time c. Typical surge waveform impulse generator Figure 1.5: Waveshape of lightning strokes.(Ref.5) Figure 1.6: World Distribution of Thunderstorm Days (Ref.6) 12 5. Frequency of lightning strokes to power lines For estimating the number of lightning strokes to power lines, we can start, from the 'electrical shadow' cast on the ground by the tall tower structure with power lines. The frequency of lightning strokes on the 'electrical shadow' is assumed to be the frequency of strokes to the power lines. The width (w) of the shadow area estimated by reference 6 is chosen. For a power line with two ground wires, the width is given by (see Figure 1.7) w = 4h + b where h = height of ground wire in m b = separation between ground wires Similarly, for a power line with only 1 ground wire, the width is given by w = 4h where h = height of ground wire in m and for power lines without ground wires, the width is given by w = 4h + b where h = height of phase wire in m b = separation between outermost phase wires Thus, the number of strokes/km - yr to the power line (N ) is Li NL = 0.1 (IKL) stroke /km -yr (1.1) 13 For a typical 500 kV tower of the MICA Dam Project, for the line close to the substation, we have AT fr\ iwor>\ 4 x 37.5 + 18.64 L = (O-1^30) loOO (1'2) = 0.5 strokes /km - yr GW GW GW = ground wire M. O : PW PW = phase wire 2h b 2h $s = shielding angle = 23° h = height of ground wire = 37.5 m b = width between ground wire = 18.64 m Figure 1.7: Lightning stroke 'Electrical shadows' of a typical 500 kV transmission line. 6. Shielding failure phenomenon of lightning strokes As shown in Figure 1.7, ground wires are designed for shielding of the phase wire from direct lightning strokes. However, lightning strokes 14 could still 'sneak' through the ground wire and hit the phase wire. Such shielding failures have been recorded in various countries for different tower conf igurat ions. Maikopar^ derived a shielding failure curve based on observed field data (see Figure 1.8). However, the graph does not shows the fact that shielding failures occur mainly on lower lightning strike currents. At higher currents, (e.g. >14.2;kA for MICA), the phase wire is effectively shielded from lightning strokes. As seen from Figure 1.1 and 1.2, the pilot downward stepped leader from the thundercloud is formed and propagates earthward freely regardless of the structure on earth initially. Later, the return strokes is formed from a ground object closest to the leader tip, (ground wire, phase wire, or the ground) arid propagates upward to meet the stepped leader to complete the lightning path. This ground object is the object which will be struck by the lightning stroke. 8 Brown analysed results from the 120,000 km - yr line in the Path finder Project and deduced that the target is not chosen until the distance between the stepped leader tip and the prospective object is shorter than the striking distance r . This striking distance is related only to the stroke current as 7 i T 0 • 7 5 rg = 7.1 I m where I = current in kA From this striking distance concept, we can develop the electro-geometric model, as shown in Figure 1.9a. The shielding failure of ground-wire at lower current amplitudes can correctly explained by this more refined method. The degree of exposure of different conductors is 15 SHIELD ANGLE - DEGREES Figure 1.8: Probability of Shielding Failure vs. Shield Angle between Ground Wire and Top Phase Conductor/ represented by drawing exposure arcs of striking distance radius, and centred at each individual conductors. The initial power frequency voltage of the phase conductor is ignored as this voltage is comparatively small to the discharge voltage of the lightning strokes. For lightning currents of 10 kA and 14.2 kA, the corresponding exposure of the phase conductor PW is shown. It can be seen that the phase conductor exposure to lightning stroke is decreased with increases in striking current. For current of amplitudes higher than 14.2 kA, for this tower structure in MICA project, the phase conductor is effectively shielded by the ground wire and the ground as the exposure arc is negligible in size (See Figure 1.9a). 16 BC=phase wire exposed arc for shielding failure SI Figure 1.9a: Electrogeometric model with maximum striking distance of 53.3m. Stroke Current in kA Figure 1.9b: Frequency Distribution of Shielding Failure Stroke Currents in case of Shielding Failure 17 Brown et al further investigated this situation by taking the angular distribution of the lightning stroke g(^) into account and evaluated the phase wire exposed arc for different stroke currents as x = r h sinii^l ^ 2 2 where: g(^) = - cos y (1.4) our The detailed analytical results are shown in Figure 1.9b. For MICA tower of maximum striking distance of 53.3 m (I =14.2kA), the result shows that less than 1% of shielding failure lightning currents to the phase wire will exceed 14 kA. This agrees well with the geometric interpretation shown in Figure 1.9a. The lightning stroke usually hit the ground wire or the tower. In this case, a voltage will build up ."across the insulator because of the potential rise on the tower crossarms. If the insulator flash over (' backflashover') to the phase conductors, then lightning overvoltage surges will appear on the conductors. 18 CHAPTER 2 : LIGHTNING SURGE PROPAGATION IN OVERHEAD TRANSMISSION LINES 1. Introduct ion Propagation of lightning surges due to direct strokes, or backflash-overs in overhead lines influences the choice of insulation require ments. One must know the attenuation and distortion characteristics of the line in order to find the overvoltages entering the substation where most of the equipment is concentrated. This section tries to answer the questions whether it is possible to represent untransposed overhead lines as equivalent single phase lines for the stricken conductor with accuracy, and whether self, positive or zero sequence impedances should be used in such single-phase representations? At first, field tests results are duplicated by using a Fourier 9 transformation method. This method not only includes the frequency-dependence of the line parameters, but it also uses the exact complex, frequency-dependent transformation matrix which requires recomputation at each frequency within the frequency range.typical of lightning surges (e.g. 10 k Hz to 1MHz). This method is recommended for the simulation of distant strokes where the frequency dependent characteristics must be included. For close-by lightning strokes, the above frequency-domain solution can be replaced by a simpler time-domain solution method. This method is based on modal analysis with frequency-independent parameters and real-valued transformation matrices. The results obtained with the simpler time-domain simulation method agree very well ( < 4% deviation) with the accurate frequency-domain simulation method. After confirming the correctness in the time-domain simulation with the exact N - phase representation of the overhead line for close-by lightning strokes, the results obtained are thus compared against single-phase approximate representations as presently used. Furthermore, additional recommendations are made on how to remove uncertainties in the choice of surge impedance values of overhead lines.^ It is also found that frequency dependence effect of nearby lightning stroke can be ignored, Line parameters can be chosen at high frequency e.g. at 1 M Hz, and line 11 12 resistance can be ignored as contradictory to the previous findings. ' 2. Modal analysis for N - phase untransposed line The well known transmission line equations describe the propagation of electromagnetic waves on overhead transmission lines. However, contrary to the single phase case, the solution to the N- phase case cannot be obtained easily since each of the N overhead conductors is mutually coupled to the other conductors. The following two sets of simultaneous second-order partial differential matrix equations describing the change in voltages and currents along the N - phase line must be solved: dVJ phase" dx nxl r phase r phase " 1 JnxnL Jnxl (2.1) dl dx phase" nxl = rYPhasei rvphasei L JnxnL Jnxl (2.2) where [zphase]. [Yphase-| [-,-phase j nxn nxn nxl [V phase. Jnxl impedance matrix in phase domain admittance matrix in phase domain phase current vector phase voltage vector 20 The N coupled differential equations in equations (2.1) and (2.2) transformed into N decoupled equations by replacing phase quan tities with modal quantities, [vphase] = [T ] [vmode] (2.3) [Iphase] = [T_3 [Imode] (2.4) and by choosing [T ] and [T±] in a certain way, as described later. Applying equations (2.3) and (2.4) to equations (2.1) and (2.2) gives demode dx [T,]"1 [zPhaSe] [T.] [Im°de] (2.5) _ j-^mode^ |.^.mode-| (2.6) and dl dx mode = [I.]"1 [YPhaSe] [T ] [.Vnode] (2.7) = j-Ymode^ ^ode^ (2.8) To find x, and replace ^T ], we first differentiate equation (2.1) with respect to *" V phase] with equation (2.2): dl1 dx dVhasel dx2 ^phase-j j.Yphase^ ^phase^ (2.9) With equation (2.3), this can be written in modal quantities as 2„mode dx = ty"1 [Zphase] [YphaSe] [Tv] 1^°**] (2.10) (2.11) If [T 1 is the matrix of eigenvectors of IZphaSe] [yphase], then [A] v becomes a diagonal matrix, with its elements being the eigenvalues of j-zphase^ ^phase^ Similarly, for the current quantities, we have J2Tmode a L_ L dx2 = [T.]-1 [YPHAS£] [ZphaSe] [T.] [Im°de] (2.12) - UJ [Im°de] (2.13) where [T\] = matrix of eigenvectors of [Y*\ ] [7? ], with [A] being identical to that in equation (2.11). Taking the transpose of the expression for [A] in equation (2.12) and comparing it with that for [A] in equation (2.10), while remembering that [Zp^ase] and [yP^ase] are symmetric, gives: [A] = [?±f [ZphaSe] [YPHAS£] ([T.]^"1 = ITv]_1 tZPhaS£] [YPHASE] [TV] or [Tv] = (-[T.]1")"1 (2.14) Thus, only one of the matrices or [T ] is needed. Using only , rm -i • i . i J i <r rr,niode, . r,,mode, the [T^J-matrix, we can obtain the modal parameters of [Z J and [Y J from equation (2.6) as [Zm°de] = [7±f [ZphaSe] [T.] (2.15) and from equation (2. 8) as [YM°DE] = [I.]-1 [YPhaSe] ([T.]11)-1 (2.15a) r mode,-! _ t rvphase,-l rm , or [Y ] = [T.] [Y ] [T.] (2.16) 13 14 In the computer pugram developed for this modal analysis, ' equation (2.16) is used for these two reasons: it does not require the inverse of [T ] and secondly, the program calculates [Y ] first anyhow, from which [Ypkase] is obtained by inversion. [Ymode] is then easily obtained by taking the reciprocal of the diagonal elements of the right-hand side of equation (2.16). [Zm°de] is not calculated from equation (2.15), but in a simpler way from [Zm°de] = [A] [Ymode]-^ (2.17) that is, each component is simply „mode l i Ymode (2<lg) This is valid because [A] from equation (2.11) can be rewritten as IA] = [T^"1 [ZPhase] [YPhaSe] [TJ . = [T,]"1 [ZphaSe] [T.]. [T.]"1 [YPhaSe] [TJ = [ZmodeJ. [Ym°de] (2.19) 3. Rotation of eigenvectors for zero shunt conductance It has to be noted that the eigenvectors (columns of [T^] or [Ty]) are only determined to within a multiplicative constant. Each eigenvector can, therefore, be multiplied with any non-zero complex scalar, and it will still be the correct eigenvector. Since we assume zero phase shunt conductances (corona losses will be discussed later in Chapter 4), the modal conductances should also be zero. This can be achieved by multiplying the eigenvectors with a properly chosen constant. Then equation (2.8), which is defined in the frequency domain, 23 can be rewritten in the time domain as follows 8.mode 1 3x = [cmode] 3 modd v 9t In order tp obtain zero modal conductances, a rotation scheme is used which makes the modal admittance matrix [Ymoc^e] purely imaginary, mode = { } j[Bm°de] ^ L Jrotate L J JL Jrotate This rotation is equivalent to dividing the i-th eigenvector (i-th column of [T ]) by a factor D^. First, find the angle 6^ of Y^mode, as shown in Figure 2.1. Then 90° - 0. D. = e I (2.20) With all Dj^'s forming a diagonal matrix [d], the modified matrix of eigenvectors becomes IT.] = [T.] [D] 1 iJrotate • iJ L J (2.21) Then from equation (2.15a), [Ym°de]rotate = [D] [T.]"1 [YPhaSe] ([T.]1)"1 [D] (2.22) or [Ym°de]rotate = [D] [Ym°de] [D] (2.23) Since all matrices in equation (2.23) are diagonal, equation (2.23) is simply a totation of Ym°de by an angle (90° - 6^, which according to „ , , r„mode, , Figure 2.1, makes [Y ^rotate Purely imaginary. 24 After [Ym°de0 is found from equation (2.23), and [T.] 1 rotate H ' 1 i rotate from equation (2.21), [Zm°de] is calculated from n rotate mode [Ymode -1 rotate rotate These modal quantities and transformation matrices obtained are characteristics of the particular design of the untransposed line. These modal parameters and modal transformation matrices are needed as input for the representation of untransposed distributed - parameter lines in the time domain solution, such as in the UBC version of the Electro-13 14 magnetic Transients Program as described in ' Figure 2.1: Complex Ymode before and after rotation 25 4. Confirmation of accuracy of eigenvalue and eigenvector subroutine The UBC Computing Centre library subroutine DCEIGN^^ is chosen to compute the eigenvalues and engenvectors of the [Y]-[Z] matrix. This double precision subroutine first reduces the complex matrix to a Hessenburg matrix H. The subdiagonal elements of H are then forced to converge to 49 zero by the modified LR method. Hence the diagonal elements of H converge to the eigenvalues. The eigenvectors can then be obtained by backward substitution. The correctness of the program has been checked by comparing its output with published results for a double-circuit line"*"^. Both results of modal attenuations and modal velocities agree to within three digits (see Table 2.1). The modal matrices [T ] differ only slightly (see Table 2.2) . Table 2.1 UBC & BPA modal analysis results for a 735 kV line 16 Modal attenuation .15998E6 .18438E6 .18497E6 .18606E6 .18615E6 .18614E6 neper/mile BPA .15998E6 .18437E6 .18497E6 .18605E6 .18614E6 .18614E6 Modal velocity UBC .61227E-1 .19050E-2 .18209E-2 .54529E-3 .50169E-3 .47704E-3 miles c/s BPA .612E-1 .191E-2 .182 E-2 .544E-3 .502E-2 .475E-3 5. Real-valued frequency - independent transformation matrix Time domain solutions with the transformation matrix [T_^] become difficult in theory since [T^] is complex as well as frequency- dependent. Table 2.2 UBC and BPA modal matrix [T ] results for a 735 kV line . 3412-j.0022 .3948-j.0157 ,4822-j.0 . 3412-j.0022 .3948-j.0157 .4822-j.0 . 3412-j.0022 .3955-J.0157 .4832-j.0 .3412-j.0022 .3955-j.0157 .4832-j.0 .5558+j.0 •3324+j.0230 -.3128+J.0248 .5558+j.0 .3324+j.0230 -.3128+J.0248 •5558+j.0612 .3294+J.0594 -.3153-j.0095 •5558+j.0612 .3294+j.0594 -.3153-j.0095 -.4959-j.0262 .5486-j.0 -.1118+J.0056 -.4959-j.0262 .5486-j.0 -.1118+j.0056 -.4959+j.0452 .5453-j.0784 -.1105+j.0215 -.4959+j.0452 .5453-j.0784 -.1105+j.0215 .1730-j.0017 .4647-j.0247 .5410-j.0 -.1730+J.0017 -. 4647+j.0247 -,5410+j.0 .1730-j.0298 .4681-J.0598 .5469-j.0408 -.1730+J.0298 -.4681+j.0598 -.5469+j.0408 .3209+j.0008 .4804+j.O -.4145+j.0304 -.3209-j.0008 -,4804+j.0 .4145-J.0304 .3209+j.0208 .4659+j.0285 -.4064+J.0048 -.3209-j.0208 -.4659-j.0285 .4064-j.0048 .6827+j.0 -.3550-j.0021 •0812+j.0061 -.6827+j.0 .3550+j.0021 -.0812-j.0061 .6827+j.0737 -.3637-j.0422 .0869+j.0159 -.6827+j.0737 •3637-j.0422 -.0869+j.0159 27 However, the imaginary part of the matrix [T ] is always small 5%) compared with its real part. By taking the real part or the magnitude of the matrix itself, we obtain modal parameters which-are • still accurate enough ( ^ 2% deviation). Furthermore, the attenuation caused by corona may be much higher than that caused by the series resistance and for close-by strokes, trans mission lines should be represented as lossless. With the approximations, the frequency dependence of the modal transformation matrix disappears. It is therefore recommended that the complex matrix be approximated by a real-valued, frequency-independent matrix. This makes simulations much easier for two reasons: a) A frequency independent modal matrix does not require recompu-tation of the modal matrix at each frequency considered within the lightning frequency range, e.g. 10 kHz to 100 kHz. b) A real-valued modal matrix enables direct transient simulation to be performed in the time domain. 6. Frequency dependent effects in lightning surge propagation To include frequency dependent effects in transient overvoltage 17 18 studies is a complicated topic by itself. Meyer, Dommel and Marti have investigated the t ime domain methods using convolution integrals and weighting functions. However, the frequency domain solutions can also be Q obtained by the Fourier Transformation methods. Though the frequency domain method is inadequate to account for the non-linear phenomenon (e.g. corona discharge) and the time domain phenomena (e.g. insulator back — flashover or arrester operation), it is sufficient for the purpose 28 of studying frequency dependent effects on lightning surge propagation in over head lines. 9 As discussed in an earlier work , the frequency domain solutions includes frequency dependence of line parameters. It also uses the exact complex frequency dependent transformation matrix to be computed at each frequency point, and employs the linear interpolation technique in evaluating the Fourier Transformation integrals. The results from a measured field test 19 20 by Ametani ' of a laboratory generated distant lightning wavefront 83.212 km from the substation was successfully duplicated by the author using the Fourier Transformation method. (See Figure 2.2). Due to the frequency de pendent effect of the line parameters, an initial rise time of 2 ys of the wavefront now increased to about 40 ys as the wave travelled down the line. Thus, the frequency dependent effect must be included for th§ distant lightning stroke case. The lightning waveshape obtained after the stroke has travelled from the striking point to the substation can then be interfaced with the time domain solutions as used in an electro-47 magnetic transients program. For close-by lightning strokes, the resulting waveshapes can again be obtained by the Fourier Transformation integrals, and the simpler time domain methods. For the time domain method,- the multi-phase untransposed line can be first solved by modal analysis using frequency independent parameters and real-valued transformation matrix (as described in previous sections). Then, this multi-phase line is represented by a single phase line approximation. As shown in Figure 2.3, results obtained by all these methods agree quite well (< 4% deviation). The single phase line representation with frequency independent effect is valid in this close-by stroke case because variations among the modal arrival times at range of lightning frequencies are not apparent in such short distances (e.g. < 2 km). Voltages(p.u) 3<|> with frequency dependence 3<j> without frequency dependence without frequency dependence ime Figure 2.3: Close-by lightning stroke case solved by frequency and time domain methods. 30 t=0 83.212 km A B • C Output voltage (p.u.) 1.0 0.8 0.6 •0.4 0.2 -0.2 field measurements 3<j> with frequency dependence Figure 2.2: Numerical simulation of over voltage taking untransposition and frequency dependence into account.(Ref. 9) 31 In such a representation, self impedance - parameters calculated at higher frequencies (e.g. 1 MHz) should be used to approximate the frequency dependance characteristics of the line. However, caution must be taken in choosing line resistance for the lightning surge studies. The frequency dependence of line parameters of one phase for a typical 500 kV line is shown in Table 2.3. It is shown that the attenuation of the wave is negligible (< 5%) up to about 100 kHz for 1 km. The resistance to reactance ratio is also small especially at higher frequencies, e.g. 2.8% at 1 M Hz. Furthermore, since the Bergeron's method of characteristic in solving the transmission line equation is valid only for a lossless transmission line, distributed line losses are usually approximated by lumping the resistance at certain locations. This high resistance at 1 M Hz may cause inaccuracy in the simulation. On the other hand, surge impedances calculated by zsurge= A + J^L (2>25) where R/jwL = 2.8% at 1 M Hz (See Table 2.3) zsurge= fcjL (2>26) / JUJC are essentially identical for this lossy and lossless cases. Thus, com plicated frequency dependent effects for the nearby stroke case can be ignored, and . frequency independent and lossless representation give acceptable accuracy(.,,See Figure 2.9 ). ./ . Therefore,.the previous methods of modelling line losses by simple • -i j • 12 • i 11,21 exponential decay in overvoltages or any resistance lumping scheme are not acceptable. They should be replaced by detailed weighting function techniques, or Fourier Transformation methods for distant stroke, or by 32 Frequency Resistance Reactance R/X (Hz) R(fi/km) X(ft/km) % zsurge velocity Attenuation ( a ) (m/ys) e"Y^( /km) in6 183. 6525. 2.8 291. 280. .73 io5 42. 692. 6. 300. 272. .93 io4 7. 78. 9. 317. 257. .989 io3 .9 8.9 10. 340. 240. .999 Table 2.3: Frequency dependence of self quantities of line parameters for a 3 phase 500 kV line. 33 lossless line representation for a nearby stroke as described in above, 7. Determination of the surge impedance of the struck phase of a  transmission line. An accurate and reliable value of the surge impedance in phase domain must be obtained as due to the following reasons: a) The amount of overvoltage wave transmitted from the overhead line to the underground SF^ cable at the cable-line junction is determined by the surge impedances of different components. The refraction coefficienttC_ is R cable" CR , 2 Zsurge 7) + ^cable surge surge where ^cable _ gu e impedance of cable (e.g. 60 ft) surge zline _ gur impedance of line (e.g. 304 ft) surge b) The exact value of the overvoltage wave on the line, resulting from the lightning stroke (1^) is directly related to surge -i . r,line impedance of the lxne Z as r surgv = ^ . zline (2.28) 2 surge This resulting overvoltage wave impresses electrical stress on external and internal insulation of the system and forms the main concern in the insulation co-ordination study. 34 im In spite of the above important criteria, uncertainties in surge pedance calculations of overhead line do exist. ^'^'^ Reference 11 give relatively lower surge impedance results for the ground wire (352 ft) 12 as compared to Darveniza's computation. Darveniza claims that the equation for surge impedance in phase domain as is given by: ZSU*f = 60 In ^ " (2.29) self r ZSUrg€\ = 60 In ^- (2.30) mutual b.. where h = conductor height ;r = conductor radius a. .. = separation between conductors ij b. . = separation between conductor and ij other conductor image This is readily derived from the potential coefficient P and the inductance term L as: 5seif • hrr ^r- "ln T <2-3I> Jself L 1 = Ho- to ^h (2-32) self 2TT r and Z^f = /^--- A __P = 60 £n^ (2.33) self / C ... self self ~ self 35 where u = permeability ° -7 = 4TT x 10 H/m e = permit ivity ° 1 -9 = x 10 F/m 36TT [C] = [P]_1 [P] = potential coefficient matrix, with diagonal term P self [L] = inductance matrix, with diagonal term L ° self However, the above formulae neglect • Carson's correction terms, other conductors, and ground wires used for earth return. A detailed calculation for the surge impedance matrix in phase domain [Z ] & r v surge must be performed to in order to justify this assumption. If we consider the relationship between the surge impedance matrix in both phase and modal domain as [yphase] = [zphase][;Iphase] (2_34) and [Vm°de] = [Zm°de ][ Im°de] (2.35) surge then by substituting eqs.(2.3) & (2.4) into (2.35), we can get [T TV1**36] = [Zm°de ][T ]_1[lphaSe] (2.36) v surge I or [VphaSe] = [T ][Zm°de ][T.]-1[lphaSe] (2.37) v surge I 36 Comparing equations (34) and (37) , we thus obtain [Z phase surge ] = [T ][Z .mode 'surge J[T.] -1 (2.38) The above relation in equation (2.38) is identical to that derived by Wedepohl. 22 In his method, the reflection coefficient for phase current is first obtained. The coefficient is then set to zero to obtain the expression for [ZP^ase] as in equation (2.38). surge M Results for the surge impedance from equations (2.29) and (2.38) for both the ground and the phase wires are shown in Table 2.4. As can be seen from the table , the surge impedance obtained by Darveniza's formula which neglects the skin effect of the earth return component introduces negligible deviation (about 1%). However, the Darveniza's formula should only be used when the ground wire is treated as another individual phase (e.g. for the close-by lightning stroke case). If one takes the ground wire as another component for earth return (e.g. for the distant stroke case), the formula for self surge impedance must be modified accordingly by treating voltages on ground wire to be zero. This requires reducing the impedance and admittance matrices before surge impedances can be calculated. The surge impedance value obtained in this case is lower than that obtained by Equation (2.29), as shown in Table 2.4. 8. Single phase representation for close-by strokes on double circuited line After the author has verified that single phase representation with appropriate choice of line parameters is accurate for a three phase line case without ground wire r a double-circuited overhead transmission line 23 of the MICA Project of the B.C. Hydro and Power Authority was used as a more detailed transmission system with ground wires. 37 surge Impedances ground wire phase wire A 547 ft 342 ft B or C 545 ft 338 ft D - 318 ft (A-B)/A*100% 0.4% 1.2% A = Exact method (2.38) with Carson.':s Correction terms for earth return skin effect, ground wire treated as another phase. B = Exact method equation (2.38) without Carson's Correction terms for earth return skin effect. C = Darveniza approximate equation (2.29). D = Exact method equation (2.38) with Carson's Correction terms for earth return skin effect, ground wire treated as earth return component. Table 2.4: Self surge impedances for ground and ^7 phase wires for a typical 500 kV line ' 38 This tranmission system is a double-circuited 500 kV line. Each tower consists of a three phase line with two ground wires (See Figures 2 .4 and 2.5). When the lightning stroke hits either one of the ground wires or one of the phase conductors, different line parameters must be chosen because of different line design. The corresponding parameters are shown in Table 2.5. One can see from this table that the self surge impedance of the ground wire is greater than that of the phase conductor. The wave propagation velocity is also lower in the ground wire case. In Figures 2.6 and 2.7, one can compare the lightning overvoltage wave propagation characteristics for the open and short circuit test by using multi- and single-phase line representation when lightning stroke, hits the ground wire. Figure 2.6a shows the result obtained by the multi phase solution method using modal analysis. It also shows clearly the different modal components on the resulting waveform. Figure 2.6b shows the result for the single phase case and the overall important propagation charactertistics of multi-phase representation is successfully duplicated here. Similar results are obtained for the short-circuit test, as shown in Figures 2.7a and 2.7b. One can observe that the current waveforms obtained from these two different line representations agree very well. Similarly, the open and short circuit test results are also successfully duplicated for surges on the phase conductor as in case of direct strokes or backf lashover s ( See Fig.2.8&2.9) Thus, it is recommended to use single phase representation for double-circuited line with ground wires for studying close-by lightning stroke propagations. 23 unit 5-3/4x10" shielding ground tower insulator //wires extending strings // 1.6km beyond substn 3 phase conduc tors tower Typical lightning arrester characteristics: nominal rating (reseal voltage), switching sparkover Min 60 Hz sparkover lightning sparkover 3 phase underground SF- cables 6 -lightning arrester Jat transformer ^jO^L-^ cable junction .transformer to be protected 420 kV 950 kV 568 kV 985 kV 396 kV 960 kV-555 kV 990 kV-Insulation levels 60 Hz BIL SF, cable Transformer 6 800 kV 745 kV 1550 kV 1675 kV Figure 2.4: Layout of SFg substation protection scheme showing one of the double circuit systems. 40 10 — 9 4 1 5 8 7 6 12 3 Conductors 1-3, 6-8 phase wires 4,5, 9,10 ground wires Coupling factor: lightning struck <j) - wire K 34 ^phase surge 3,4 ^phase surge 3,3 •16 y induced ground wire V phase wire lightning struck g - wire K ^phase _ surge 4,3 _ 43 phase Jsurge 4,4 •06 induced phase wire V ground wire Figure 2.5: Side view of the MICA 10 <j> Systems. Ground conductor Phase conductor Self surge impedance Z^|e 658 ft 304 ft Wave velocity v 245 ™/\is 293 m/ys Line resistance 0 ^/m 0 ^/m Length 1609 m 1609 m where ZS^f = 60 In ^ self r = /L P 77 self self velocity = self  Lself and L . , _. and P . , , are diagonal elements of self S 21 matrix [L] and [P]. Table 2.4: Line parameters of ground and phase conductor for lossless single phase representation. 42 Different modal arrival times Single phase arrival In 1<|> case: VT2p3 = (Coupling factor) surge = 3'4 V surge T2G4 Z4,4 Figure 2.6: Open circuit test on single and multi-phase representation with stroke on ground wire. 4 3 i Different modal arrival times Figure 2.7: Short circuit test on single and multi phase representation with stroke on ground wire. Figure 2.8: Open circuit test on single and multi-phase representation with stroke on phase conductors. 45 Small difference in different modal arrival times _3 iT2p3(10 P-u.) t-0- \. 304« TP3 T2P3 1+ 4 2 2 p.u. (S T2P3 Time t} ii,.or 4 f 8 12- 16 (ys) Single phase arrival Figure 2.9: Short circuit test on single and multi-phase representation with stroke on phase conductors. 46 CHAPTER 3: LIGHTNING WAVE PROPAGATION IN SF6 GAS INSULATED UNDERGROUND TRANSMISSION CABLE SYSTEM. 1. Introduction The world's first commercial SF, gas -insulated cable rated at 6 345 kV was installed in 1970. Its inherent advantages over conventional underground oil-filled cables with respect to charging current, dielectric losses, thermal performance, voltage rating flexibility and power handling capacity are well-known. It offers additional advantages of reduced substation size. This compactness in size of SF^-insulated substations and switchgear brings the equipment closer to the protective lightning arrester located at the overhead line and underground cable junction. This is of.vital-importance, especially when there is no lightning arrester at the transformer terminal, as in certain substation design. 23 In the SF,-insulated cable at the MICA Dam, which will be used o as a test example, each of the 3 phase cables consists of two concentric aluminum tubes (see Figure 3.1a). The inner tube is the conductor core and the outer grounded tube is the sheath. The three sheaths are solidly bonded together and grounded at many locations. At the high frequencies encountered in lightning surges, the sheath return current will be equal in magnitude and 180° out of phase with the core conductor current. Whether the magnetic field external to the sheaths can be completely neglected in the frequency range of interest must be investigated, however. If the magnetic field is negligible, then there would be no mutual inductive coupling among the phases. There is no electrostatic capacitive coupling between phases as the solidly-grounded sheaths act as electrostatic shields Up to 1 Mhz. 47 Skin depth of Al, •t * \-1/2 8.1 = (.TTfau) = cm =1.0 cm at 60Hz Scale--= 1:2.54 r = r„ = 3" = 7.62 cm 3.5" = 8.89 cm 9.75" = 24.765 cm r. = 10" = 25.4 cm 4 sheath thickness=.635 cm Permeability Al = y u, = u-r 0 0 = 4TT .x 10 -7 H/ m Figure 3.1a: Individual cable design Permittivity SF& = £r £Q = eQ 1 x 10 9 F/m 36TT Figure 3.1b: Overall cable layout. 48 In order to investigate the sheath current return phenomenon of SF, b cables, and thus to clarify the wave propagation characteristics, the cable parameter must be calculated accurately. In this research work, cable 24 25 parameters of multi-core cable ' is not investigated. Such multi-core cable systems certainly have coupling between phases at all frequencies. Nevertheless, coupling between phases for the single core cable system in different frequencies needs further investigation. 26 27 Commellini and Abledu used finite elements technique to sub divide the main conductors into smaller sub-conductors of cylindrical shape. The impedance matrix for the main conductors was formed by bundling up the sub-conductors in the matrix elimination process. However, due to the thin tubular shape of the conductors involved in the SF^ buses, large number of sub-conductors is required. This will demand huge computer core storage space and long computer execution time. 28 29 Sunde and Pollaczek had derived analytical expressions for the self and mutual impedance of cables which are constructed overhead, underground or on ground surface. These analytical expressions contain Kelvin functions and an infinite integral known as the Carson's Correction terms. Before the widespread application of digital computers, simplified assumptions and restrictions were made to facilitate the computation process. With the recent popularity and increased application of digital computers, these infinite integrals can be modified and replaced by straight forward numerical computations without significant sacrifice for accuracy. 30 31 32 Wedepohl et al and Ametani ' used different approaches to tackle the analytical expression in the cable parameter calculation. However, both approaches gave different results (20% from each other). Bianchi proposed to calculate the earth or sea return impedance by approximating the return medium as a tube of infinite outside radius. These approximation results fell somewhere between those of Wedepohl et al and Ametani. Because of the inconsistency in the above findings, a detailed investigation for numerical calculation of cable parameters must be performed to reveal wave propagation characteristics in SF^ single core cables. Current return characteristics from core through sheath also must be investigated to confirm single phase or multi-phase representation for the cable system involved. 2. Formation of series impedance matrix for SFfi cables The single core SF^ cable system configuration is shown in Figure 3. Each phase consists of two conductors, core and sheath. We can build up a 6 x 6 series impedance matrix Z, describing the cable system as follows: dVcln dx z ii sll Z 19 sl2 Zml2 Zml2 Zml3 Zml3 " ^l" dV . si dx Z 01 s21 Zs22 Zml2 Zml2 Zml3 Zml3 ^1 dVc2 dx Zml2 Zml2 z ii sll Zsl2 3nl2 Zml2 ^2 dV s2 dx Zml2 Zml2 Zs21 Zs22 Zml2 Zml2^ • ^2 dV _ c3 dx Zml3 Zml3 Zml2 Zml2 Zsll Zsl2 ^3 dV _ s3 dx _ Zml3 Zml3 Zml2 Zml2 Zs21 Zs22 _ . ^3. 50 z s Zml2 Z, " ml3 ^1 Zml2 Z s Zml2 \l ^2 (3.1) Zml3' T Zml2 Z s ^3 s where Z is... 'self submatrix on the diagonal. All Z matrices are s ° s equal because they represent identical cable configurations. It is assumed: that the mutual impedance between cores, between sheaths and between corresponding cores and sheaths are all equal. In other words, all elements in the sub-matrix Z , „ or Z. are ^assumed ml2 ml3 to,; be equall:J:-(Bee Section 6 for "further discuss ion) • — 3. Calculation of self and mutual earth return impedance  for single core cable The analytical expressions for the self and mutual earth return 29 impedance of cables was first derived by Sunde and Pollaczek^ and then by Wedepohl et a£? Firstly,the Maxwell's electromagnetic equation can be solved after neglecting end effects as: 51 V x E = -ja>y0 H (3.2) V x H = J + |2. = j ( i + (3.3) 3t a = J, as displacement current , can be neglected. Taking the curl of equation (3.2) and substituting into (3 3) give V x V x E = -jwy0 V x H or V(VE) - V2E = -juyQ(J) Assuming cable separation >> cable radius, we have V2E = jtoy0 a(E +p i 6(x) <5(y + h)) Assuming cables are parallel to ground surface and attenuation of voltage and current is negligible over distances comparable to cable separation, 2 2 we have 3 En 3 E, + —T- = 0 » y > o. (3-4> 2 2 3x 3y 2 2 3 E„ 3 E —j- + Y = m2E2 +p m2 i 6(x) 6(y + h) , 3x 3y y < 0. (3.5) where m:; = J j cjy P ... 1 P •• = earth resxstivxty = — oj = angular frequency y = permeability 6 = Difac function E^,E2 = electric field above and below ground. Imposing the restriction of continuity of electric field at y = 0 for E^ and as boundary conditions, we can obtain a general expression 52 y 7TT Figure 3.2: Conductor configurations for earth return formula. 53 for the underground electric field . We can then obtain the expression for the mutual earth-return impedance of underground cables by dividing E2(x,h2) by the current i (O.h^) to 8et Z. . = ij 2IT 00 / 2" 2* 00 '/2 2~\ exp(-£/ a +m ) &j ax ^+ j exp;(-l//a +m ) da -1 1 ' n~~2 a + Va +m 0/2.2 2/a +m exp( -£Va2+m2) 2v42+m2 da (3.6) exp,(= v^/a2-rm2: lax 2^.00 1 a I +/a2+ra2 da + (Kn(mDj - (K0(mD0)) 2 77 s (T 1 0: 2' = A Z4J + 4^ fe-JuDj - K„(mD„)) Jij 1 277 ^o^l CT 2y (3.7) where = Kelvin function of order zero m D, = = y^^" jx = horizontal separation between cables /x /2 2 D2 = /x +(^4-^) h^,h2 = depths of burial of cables a = l = |h1 + h I AZ.. = Carson's Correction term, identical to that for over-head lines case. The above formula is also applicable to self earth return impedance components. In such cases, the terms for and can be re-defined as D1= r D2 = 2h (3.7a) • where r is radius, h = depth of buried cable. However, the above formula is unsuitable for straight forward calculations. Before digital computers are widely used, approximate results were obtained 28 only after certain limiting conditions were accommodated. Then, Wedepohl et al used Cauchy's integration for the Carson's correction terms and derived approximate formula for the equation given in equation (3.7) as mD „ Z. . = ^ { £n(Y-±-) + i - f mA } (3.8) ij • 2 ir 2 2 i for |mD | < 0.25 and Y = Euler;'s constant = 0.5772157. However, the above formula gives results which are about 20% higher when compared with the direct numerical computation using the original equation as in Ametarii's case. The infinite integral for the underground cable is the same as the Carson's correction terms for overhead lines. We can define a new parameter a as a=/^D = 4TT /5 10 4 D/|- (3.9) 55 where D and p are in MKS units 2h. for self earth return impedance ^ ~ for mutual earth return impedance This correction term integral can be represented by the following . .„ . .34 infinite converging series: 1. For a < 5 AR' = 4o)-10~4{^-o -b^a* cost)) 2 2 +b^[ (c2~lna)a cos2<{)+(j)a sin2cj)] 3 +b^a cos3<}> -d^a4cos4cf> -b^a^cos5<j> 6 6 +b^ [ (c^-lna)a cos6<j)+cba sin6c))] +b^a^cos7<j> g -dQa cos8<f> o - ...} AX' = 4to-10 4{|<0.6159315-lna) +b^a*cos<j> 2 -d^a cos2cf> 3 +b^a cos3<j> 4 4 -b^[(c^-lna)a cos4cf>+(j>a sin4cf>] +b^a^cos5cj) -d^a^cos6tj) +b^a cos7<j> 8 8 -bQ[ (cQ-lna)a cos8<J>+c(>a sin8c()] o o + ...} 56 Notice that each 4 successive terms form a repetitive pattern. The co efficients b., c. and d. are obtained from the recursive formulas: 11 l /2 ^ b^ = —-r for odd subscripts, b. = b. „ . z^foN with the starting value -\ l i-2 i(i+2) 1 ^•b„ = f°r even subscripts, 2 ID c = c. „ + + -rrr with the starting value c„ = 1.3659315, l i-2 l i+2 - 2 A 71 U d. = j- • b., l 4 l with sign = ±1 changing after each 4 successive terms (sign = ±1 for i 1,2,3,4; sign = -1 for i = 5,6,7,8 etc.). 2. For a > 5 , _ ' cos((> /2 cos2<)) cos3<}> 3cos5<j) 45cos7(j) - ( 235" 7 ) 4a)-10 4 /I Q3.ll) -4 AX1 - ( COS1^ _ cos3<l> + 3cos5<{> + 45cos7(}> ^ # 4ai* 10 3 5 7 >=-a a a a v2 It should be noted that the correction terms will become zero when the parameter a is very big, ie. when frequency or cable distance from g.round is very large or when earth resistivity is very small. The Kelvin functions can also be calculated by another infinite 36 converging series as can be obtained from availabe source. , ''-It can also 35 be calculated by a special subroutine CBESK available from the UBC Computing Centre. After establishing the numerical formula for cable earth return impedance, the discrepancies between the results of Wedepohl et al and those of Ametani can then be clarified. The author has confirmed that accurate cable earth return impedance •can be calculated by using direct computation of infinite series sub-40 stitution for infinite integral and Kelvin function. Cable mutual impedance from Ametanis computation is acceptable though different infinite series is used for the Carson's correction terms. Identical results are obtained at least to 4 significant figures for frequencies up to 100 k Hz (See Figure 3.2). The approximate formula given by equation (3.8) on the other hand, gives results about 20% consistently higher. The author has also confirmed that the earth return impedance for underground cable can be approximated by the equivalent earth return impedance for overhead lines. The expression for earth return impedance for overhead line is D Z.. = to -==- + AZ.. , .„ ij 2TT DJ^ IJ ' (3.12) for mutual earth return impedance 2h and Z. . = to — + AZ. . , (3.13) xj 2 It GMR xj for self earth return impedance where is distance between i*"*1 and image of j1"^ conductor; is distance between i*"*1 and jt'1 conductor; • h is height of conductor above ground; GMR is geometric mean radius= radius of conductor at high fr AZ is Carson's correction term. Identical results up to 3 or 4 figures are obtained for earth resistivity of 1 to 100 Q-m up to the frequency of 100 K Hz. The 58 consistency between these two results is due to the fact that the Kelvin functions K (/j" x) = ker(x) + j kei(x).; 36 can be evaluated by the following convev^ing series for 0 < x < 8: ker x=— In (ir) berz+Jxbei x—.57721 566 1 -59.05819 744(jf/8)*+171.36272 133(x/8)8 : -60.60977 451(z/8)12+5.65539 121 (x/8)'8 -.19636 347(x/8)20+.00309 699(:r/8)24 -.00002 458(2-/8)28+« (3.14) H<ixio-8 kei*=-ln($x)beia;-fir ber a,+6.76454 936(z/8)* -142.91827 687(i/8)6+124.23569 Q50(z/8y° -21.30060 904(ar/8)14+l. 17509 064(r/8)18 -.02695 875(ar/8)22+.00029 532(x/8)29+« _(3.15) |€|<3X10-» where ber x=1 -64(as/8)4+113.77777 774(*/8)8 -32.36345 652(z/8)12+2.64191 397(a;/8)16 -.08349 609(*/8)M+.00122 552(x/8)24 -.00000 90l(x/8)28+e |«|<ixio-» (3.16) bei x=16(a;/8)2-113.77777 774(x/8)* + 72.81777 742(a/8)l0~ 10.56765 779(x/8)u + .52185 615(z/8)l8-.01l03 667(ay8)22 +.00011 346(z/8)29+e M<8X10-» Figure 3.2: Mutual impedance between outermost cables by different computation methods. 60 and x = ^f- D, (3.18) u = 4TT x 10 ^ ^m oi = frequency D = distance Di or D L 2 P = earth resistivity For frequencies up to about 1 M Hz and earth resistivity of about 100 ft-m,and cable separation or cable depth of about 1 m, the term x is relatively small as x -> 0 Then, one can rewrite equations (3.14) to (3.17) for x 0 as ber = 1 bei = 0 • • KQ(/fx) = ker x + j kei x = - £n |- x - 0.57721 -j J (3.19) Thus, for the earth return impedance as shown in equation (7), we have Z. . = AZ. . + ™ (K (mD.) - K (mD.)) xj xj 2 IT o 1 o 2 mDn - 4Z«+ if « - «• -r - - jj) -mD ( - in -f - - .57721 - jJ-)) = AZ . + ^ • in ^ , x * 0 y1J 2lT j, °1 Carson's correction term self-term which is the same as in equations (3.12) and (3.13). 61 The numerical results for the self-term component of the self and mutual earth return impedance for the underground cable obtained by the different formulae developed earlier are shown in Table 3.1. As can be seen from Table 3.1, the results obtained by these different formulae are very consistent. The final results for mutual impedance from these methods are also shown in Figure 3.3 for frequencies up to 1 M Hz. Because of the observed consistencies, the overhead line formula approximation is therefore recommended for underground cable for all frequencies up to 1 M Hz and earth resistivities above 1 ft-m. 4. Calculation of self impedance matrix for single core cable After the self and mutual impedance for earth return of under^. ground cables,.is obtained, the self impedance of individual cables can be calculated and the obtained results for different current loops can then be transformed to the required form for the impedance diagonal submatrix Z as shown in equation (3.1). s At first, one can consider the current in each of the individual cables flow in two adjacent loops as shown in Figure 3.4. Loop 1 is formed by the current flowing through the core and returning through the outside sheath. Loop 2 is formed by the current flowing through the sheath and returning through the outside earth. These two loops can be described by equation (3.21) as dvl dx Zll Z12 4 11 dv2 dx _Z21 Z22 _i2_ where Z = Z by symmetry. 62 Separation in meter Dl D2 (1,3) .889 2.189 .901+j.OOO (60Hz) .901 .895-J.020 (100Hz.) .901 (1,2) 1.778 2.676 .408+j.OOO (60Hz) .408 .403-J.018 (100Hz) .408 (self) .254 2.0 2.064+j.OOO (60Hz) 2.064 2.057-J.022 (100Hz) 2.064 where f = 60 Hz, m =/j6° X 2\lQ^ X 10"7-= .0022 /J i TT /icoy /jlOO x 2TT x 4TT X 10 ^ AQn rr f = 100 Hz, m = / ^ =/J JOO -089 /j underground cable (exact) Zij = Trf (VmV ' K0(mD2» + AZij overhead cable (approximation to above) hi " ^ *> 57 + *Z13 • - - » * 0 Table 3.1: Mutual and self earth return impedance terms as given by under ground and overhead cable formula. underground overhead Figure 3.3: Approximations of mutual impedances between underground cables by Carson's formulae. Figure 3.4: Current loops inside SF^ cable for self impedance calculation. 65 The matrix elements of equation (3.21) can be obtained by considering the individual current loop components making up the corresponding loops 1 and 2 as Z = Z + Z + Z 11 core-outside core/sheath insulation sheath-inside (3.22) Z = Z + Z 22 sheath-outside earth-inside (3.23) z = z = —z 12 21 sheath-mutual (minus sign since i and ± in different direction). (N.B. Z^ is negligible when sheath thickness » skin depth) where the individual elements are (3.24) (Zl) Z core-outside (Z2) Z core/sheath insulation (Z3) Z sheath-inside (Z4) Z sheath-outside (Z5) Z earth-inside internal impedance of core with return through outside (sheath). impedance of SF, insulation due to the o time varying magnetic field. internal impedance of sheath with return through inside (core). internal impedance of sheath with return through outside (earth). self earth return impedance, this can be calculated by equations (3.7) & (3.7a), or can also be obtained by equation (3.25) with the approximation of infinite 33 outside radius (Z6) Z sheath-mutual = mutual impedance of tubular sheath between loop 1 in inner surface and loop 2 in outer surface of sheath. 66 The individual self and mutual impedance terms can be again obtained by solving the Maxwell's equations for the coaxial conductors as similar to equations "£3.2)&(3 .3) # They are a function of frequency as derived by 37 28 Schkelkunoff and Sunde as . . , = (In(mq) K, (mr) + K„(mq) I.,(mr)) tube-inside 2iTqp 0 1 0 1 (3.25) tube-outside = 2~- (IQ(mr) KL(mq) + KQ(mr) I^mq)) (3.26) Jtube-mutual with 2Trqrp p = I1(mr) ^(mq) - I (mq) K (mr) (3.27) (3.28) where and y q r m = angular frequency = 2i:f permeability = P^QJ yr = 1 for Al outside radius of tubular conductor inside radius of tubular conductor p = d.c. resistivity Vri K0''K1 p Bessel functions Kelvin functions After obtaining the individual terms of the loop equation matrix as shown in equation (3.21), we can then obtain the diagonal sub-matrix elements by applying the following circuit conditions: V = V - v 1 c s V = V 2 s (3.30) 1 = 1 1 c (3.31) i» = i + i 2 c s (3.32) The individual loop equations of equation (3 21) then becomes dV dV -T9" + -J^ = (Zn + zi,) i + Zio 1 dx dx 11 12 c 12 s dV and — = (Z + Z,_) i + Z„„ i dx 12 22 c 22 s (3.33) (3.34) Adding equation (3.33) to (3.34) gives dV dx (Z11 + 2Z12 + Z22> \ + (Z12 + Z22) is ' (3"35) Thus, we can rewrite the self sub-matrix Z as ' s Z11+2Z12+Z22 Z12+Z22 dV c dx dV" s dx Z12 + Z22 J22 (3.36) 5. /ffheath current refragn characteristics for, usual earth As current flows along the core of the buried SF^ bus, a return path is formed on its own sheath and possibly also on the surrounding soil and adjacent sheaths. Whether all the currents will return through its own sheath depends solely on the frequencies involved. Due to the skin effect in sheath material (Aluminum), all core current will return through its own sheath for frequencies above 1 k Hz. In reality, the SF^ cable is laid on the ground surface (e.g. inside the lead shaft) or is constructed above ground and grounded at certain intervals (e.g. inside the substation). This cable location even favor more current returning through the sheath than the ground as compared to buried'cable case. One can thus investigate the limiting case with the cable buried underground. This cable location will favour least 68 core current returning through its own sheath. In order to investigate the sheath current return characteristics and therefore the mutual coupling between cables, one has to use the series impedance matrix. One can consider cases in which adjacent sheaths are either included or excluded. a. Sheath current return characteristics for single cable system For this case, one only has to consider the self diagonal submatrix of the series impedance matrix as dV dx dV dx Jsll Jsl2 Jsl2 Js22 (3.37) Since the sheaths of the three individual SF^ cables are solidly grounded at short joint intervals or laid on earth surface, or buried inside .the earth, the sheath voltages can be considered to be zero for all practical purposes. Then Equation (3.37) becomes 0 Jsl2 Z or s!2 's22 i + Z c s22 I -c At high frequencies as sheath mutual impedance is negligible^ 33) when sheath thickness greater \than skin depth( See Appendix A.). Thus, neglecting the other two sheaths, the current return characteristics of the SF,, cable through its own sheath from the core o can be calculated as in Equation (3.38). The obtained results are shown in Figure 3.4 .. For this case, essentially all the current through the core will return through its own sheath above the frequency of 10 Hz. 69 "?"sheath core z 0 sheath-mutual '''sheath """core earth _ single-cable system three-cable system: -current in one core • ' -current in three cores ®- iBl/icl ®" is2/lc2 ®" 1s3/lc3 p = lOOft-m depth = 0 or .254m log frequency 1— 10 100 lk 10k (Hz) Figure 3.4: Ratio of core current return through own sheath for single-and three-cable system. 70 In other words, the sheath acts as a perfect magnetic shield above the frequency of 10 Hz. Because of this consideration, all SF^ cables are decoupled from one another and can be represented as 3 single phase systems. It is also found that a change in depth (1 m to .254 m) of cable does not change the current return characteristics noticeably. b. Sheath return characteristics for 3-cable system Since all the 3 sheaths of the SF^ bus are solidly grounded, the current will return through all the three sheaths at lower frequencies (< 60 Hz). At higher frequencies, however, all the core current will return through its own sheath because of the skin effect on the sheath. Here, again, one can conclude that the three SF^ cables are decoupled from one another. For this case of 3-cable system, one can also consider the sheath voltages to be zero. One can substitute this condition into equation (3.1) and obtain dV " cl dx -5sii Zsl2 Zml2 Zml2 Zml3 Zml3 ^1 0 Zsl2 Zs22 Zml2 Zml2 Zml3 Zml3 ^1 dVc2 dx Zml2 Zml2 Zsll Zsl2 Zml2 Zml2 \2 0 Zml2 Zml2 Zsl2 Zs22 Zml2 Zml2 is2 dVc3 dx Z'ml3 Zml3 Zml2 Zml2 Z n sll Zsl2 c3 0 Zml3 Zml3 Zml2 Zml2 Zsl2 Zs22 s3 (3.39) (N.B. Z',„ = Z as symmetrical arrangement of cables as ml2 mz 3 in Figure ,3.1b) Equating the zero sheath voltages for the 3 cables, we have 0 = Zc10 i . + Z „ i , + Z. ..(i + i ) + Z (1 + i ) (3.40) sl2 cl s22 si ml 2 c2 s2 ml 3 c3 s3 0 = Z 10(i.-. + i ) + Z i. + Z i + Z 10(1 . + i ) (3.41) ml 2 cl si sl2 c2 s22 s2 ml2 c3 s3 0 = Z ,0(i - + i n) + Z. ,0(i „ + i 0) + Z _ i _ + Z 0„ i _ ,~ /0. ml3 cl si ml 2 c2 s2 sl2 c3 s22 s3 (3.42) If we assume phase B is energized, i.e., we assume ±cl = ic3 (3.43) ^1 = S3 (3-44) itl = ic3 = 0 (3.45) Then, substitute equations (3.43) to (3.45) into (3.40), we obtain 0 = Zc00 i + Z i + Z i + Z i S22 si ml2 s2 ml3 si m!2 c2 = (Z. ,n_ + Z ) i + Z. i + Z i - ml 3 s22 si ml2 c2 ml2 s2 Znil3 + Zs22 Xsl , 1s2 ,/0 or + = -1 (3.46 ml2 1c2 1c2 Also substitute equations (3.43) tO (3.45) into (3.41), we obtain 0 " 2Zml2 \l + Zs22 \2 + Zsl2 !!i^.i5l + !522 ^=_± (3.47) Zsl2 Xc2 Zsl2 1c2 72 By solving equations (3.46) and (3.47), we get Zml3 + Zs22 ^ml2 2Z ml2 Jsl2 Zml3 + Zs22 Jml2 2Z ai2 'sl2 -1 -1 (Z ... + Z 00)Z - 2Z ml3 s22 sl2 ml2 Jml2 Js22 Zsl2 2Zml2'Zmi2 ZB22 (Zml3 +" Zs22 ) (3.48) -1 -1 Zml3 + Zs22 ml2 2Z nil2 Jsl2 Js22 Jsl2 -Z sl2 Js22 Jsl2 2Z m!2 Zml2 + Zs22 Jml2 Zml2 Zs22(Zml3 + Zs22) (3.49) The ratio of currents in sheath to core of phase B is calculated and plotted as a function of frequency in Figure3.4. The result shows that at frequencies above 1 kHz as in lightning surges, all current through the core will return through its own sheath. Thus, one can conclude that the earth return current component is not important and therefore the mutual 73 coupling between cables can be ignored. c. Sheath current return characteristics for 3 cable  system with current in all 3 cores For a case when currents flox^ in all the three cores of the three SFg cables, the return current through sheath will change accordingly. This suggests that such situations must also be investigated to deduce the mutual coupling effect among cables. Using the same equations as derived in Equations (3.40) to (3.42), one can now put in currents in the 3 cores by assuming and ic2 - i.o IR ic3 = 1.0 /120° i = 1.0 /-120' cl ' (3.50) (3.51) • (3.52) Rewriting equations (3.40) to (3.42) as ZS22 \l + Zml2 ^2 + Zml3 ^3 = 'Zsl2 \l " Zml2 ^2 " Zml3 ^3 = h <3-53)  Zml2 \l + Zs22 S2 + V? ^3 " "Z,12 Si " Zsl2 ±c2 " Zml2 S3 = A2 (3"54)  Zml3 Si + Zml2 S2 + Zs22 S3 = "Zml3 Si " Zml2 S2 ~ Zsl2 S3 = A3 (3'55) Defining the determinant T as Zs22 Zml2 Zml3 Zml2 Zs22 Zml2 Zml3 Zml2 Zs22 Zs232 + 2Zml2 ' Zml3 " Zs22(Zml3 + 2Zm12> (3.56) 74 We can then obtain the current through the three individual sheaths as' 2 2 2 isl = AlZs22 t A3Zml2 + A2Zm12Zml3 " A3Zs22Zml3 ~ A2Zml2Zs22 " AlZml2 T (3.57) 2 2 i 2 = A2Zs22 + AlZml2 Zml3 + A3Zml2Zml3 ~ A2Zml3 ~ A3Zml2Zs22 " AlZml2Zs22 T -(3.58) 2 2 2 1S3 " A3Zs22 + A2Zrql2Zral3 + AlZral2 " AlZml3 Zs22 " A2Zml2Zs22 " A3Zml2 T (3.59) After substituting the conditions, for the 3 phase currents from equations (3.50) to (3.52) into equations (3.57) to (3.59), one can obtain the return currents through all individual sheaths. The magnitudes of the sheath currents are also shown in Figure is again confirmed here that at frequency above 60 Hz, all current flowing from core will return through its own sheath. Each core is completely shielded from the adjacent cores. Thus, the three SF^ buses are completely decoupled from one another and should therefore be represented by single phases as in the case of lightning overvoltage propagation. 6. Sheath current return characteristics for substation earth with  grounding grid network In reality in the substation, the cable sheaths are grounded inside the substation with a grounding network grid consisting of copper bars which are connected across the whole substation. These grounding copper bars serve to reduce significantly the inside earth resistivity of the substation. This suggests that the sheath current return characteristics of this reduced earth resistivity should also be investigated as a reduction 75 in earth resistivity will favor more current returning through the earth. The result for the sheath return current as a function of earth resistivity is shown in Figure 3.5. This figure shows that the sheath return current increases as the earth resistivity increases. This agrees with the manufacturers and utility companies of SFg substations who claim that current returning through sheath inside the substation is less than 75%. Based upon this criteria, a nominal earth resistivity of 0.3 x 10 ^ftm is chosen. After choosing a nominal value for earth resistivity, the sheath current return characteristic is then evaluated as a function of frequencies and depth, as shown in Figure-3.6. Fluctuations in overall sheath current results are shown in Figure 3.6. At about 1 K Hz, the sheath current is even found to be larger than the core current. This can be explained by the phasor diagram as shown in Figure 3.7. In Figure 3.7,only multi-cable systems with current in centre core are shown, but mutli-cable system with currents in all 3 cores would Kralso give identical results. The present study again confirmed that all cores are decoupled from one another above 2 k Hz even for the adverse case of significantly reduced earth resistivity inside the substation. It should be noted that the mutual impedance between cores, between sheaths and between corresponding cores and sheaths are all assumed to be equal by Wedepohl and Ametani. The shielding effect of the sheath is neglected. The validity of this assumption in cable parameter computations could be the topic of further research. It is of little concern for the purpose of this thesis. At the high frequencies encountered in lightning surge studies, core current always return completely through the sheath. In that case, the magnetic field becomes zero outside the sheath anyhow. log earth resistivity.*; 6 7 8 9 10 Figure 3.5: Ratio of core,current return through sheath at 60 Hz for different earth resistivities. 20 30 40 (10"6ft-m) 77 sheath 120 100 80 60 40 z o sheath-mutual sheath earth core single-cable system three-cable system i „/i s2 c2 (§) : depth or height = .254 m (E) : depth or height = 1 m ©: depth or height = 6 m p = 3uft-m 20 log frequency i i 1 1 • 10 100 lk 10k 100k (Hz) Figure 3.6: Ratio of core current return through own sheath for single- and three- cable system at reduced earth resistivity of 3 yft-m inside substation. 78 .1 Hz ->-i core """so ii more core current return through soil than sheath 60 Hz i core :". sheath more core current return through sheath than soil lk Hz 10k Hz -> ->-i core i core , - — — — ' 1sheath -y •+ Ii J> i 1 sheath 1 core1 """sheath "ccore •"•core """sheath + """soil Figure 3.7: Phasor diagram of current return through sheath and earth. 7• Formation of shunt admittance matrix for SFfi cables 79 For a usual 3-phase single core underground cable system, one can build a 6 x 6 shunt admittance matrix [Y ] to describe the cable system as i.e. LdxJ [Y] [V] rdid~ yl -yl 0 0 0 0 dx si "yl yl+y2 0 0 0 0 • dx dic2 0 0 yl 0 0 dx I dls2 0 0 -yl yl+y2 0 0 dx di _ c3 0 0 0 0 yl -yl dx di . s3 0 0 0 0 -yl yl+y2 dx cl si c2 V s2 c3 s3 (3.60) Notice that the off-diagonal submatrix of [Y] are all zero due to the fact that the grounded sheaths in between acts as electrostatic shield between cables. For the SF^ cable system as shown in Figure 3.1, the diagonal submatrix elements are i y.. = iwc, = io) 2TT£ '1 J 1 J o Jin r3 y2 = ^wc2 = 27re0 * Jin where y^ is admittance due to internal SF^ gas insulation, and is admittance due to external sheath insulation. (N.B. ',The external insulation is non-existent for the SFg cable.) 80 As has been confirmed by the author in previous finding for sheath current return characteristics, the core should be represented as single phase. Then, the admittance equation shown in Equation (3.60) should be reduced to di C y V = J'l C dx In — r2 yself * Vc (3.62) 8. Confirmation of numerical accuracy for cable parameter calculation  and current return ratios The numerical accuracy of the computation was confirmed when the cable parameters obtained by the developed cable constants program, 32 and by the BPA cable constant program agree consistently to more than three significant figures. 38 Then, a 500 kV submarine cable was chosen as another test example. In this case, the cable parameters for the submarine cable was first cal culated. The amount of core current returning through the sheath, armour and the sea was obtained by taking into account of zero potentials on the grounded sheath and the grounded armour. The ratios of magnitudes of core current returning through the sheath ,the armour, and the sea at 60 Hz were obtained as 14%, 87.8% and 5.6% respectively. These agreed to more 48 than two figures to the results of other findings. 9. Single phase representation parameters for multi-phase SF6 cables Since all phases of the SF^ cables are decoupled from one another, single phase cable representation for studying lightning overvoltage wave 81 propagation in SF, cable is recommended. The self admittance element (y ) " self can be calculated from the simple formula as shown in equations (3.61) and (3.62), whereas the self impedance matrix element can be calculated from equation (3.1) as dVcl Z i + Z i + Z 10(i „ + i „) + Z .,(i . + i J (3.63) — == sll cl sl2 si ml2 c2 s2 ml3 c3 s3 dx and 0 = Zs21 icl + Zs22 isl + Zml2(ic2 + i^) + Zml3(ic3 + {y^ Subtracting Equation (3.64) from (3.63), we get (Zsll " Zs21> Si + (Zsl2 * Zs22) Si dx = (zsii " W1 - TT> "Wrr" Si> <3-65> cl cl = self cl where the sheath to core current ratio can be obtained from equation (3.49) or equations (3.57) to (3.59). Because all core current returns through the sheath at high frequencies, one has in such condition Si = -1, and Z = Z i , ' sl2 s22 . . cl Substituting this into equation (3.65) or (3.37), one obtains dV dx (Zsll " Zs22> S (3'66) After having obtained the self series impedance for single phase cable as shown in Figure 3...1, one can then represent the cable by the surge surge impedance Z , and the wave propagation velocity v, as given by 82 zsurge /!self = ^ fl (3>6?) / yself v = oi /- — = 300 m/ys (3.68) / self • yself One should realize that the metallic sheath of the SF^ cable always form a very good earth return path to the cable core. The series resistance is negligible compared tp the reactance (See Fig.3,8),Thus, the SF^ cable can be taken to be lossless. It should also be noted that for such a simple go-return circuit for a coaxial cable, the inductance can be given by the simple formula^ as y r L = •—- in — (3.69) - • 2TT r2 = 0.205 VH/m Consistent results for the inductance are obtained by equation (3.65) and (3.69) for frequencies above 10 Hz. Thus, the simple formula is in equation (3.69) is recommended for inductance calculation of SF^ cable in the study of surge propagation characteristics. The surge impedance and7 the wave propagation velocity can be then obtained as surge = /1 = A° .. £n I3 . _1_ . £n I3 C y- 2TT r2 2TTEo r2 ° 1 „ 3 e , 2 r„ o 4TT 2 r3 = 60 £n — (3.70) r2 where r^ and r2 are radius of sheath and core respectively. 10 Figure 3.8: Self series inductance, resistance and resistance to reactance ratio for SF, cable. 84 and v= /f^ = /— Jin— (3.70) •' T r / y r„ 2TTE r„ r2 y £ o o = 300 m/ys velocity of light in vacuum 10. Wave propagation in SF6 cables Wave propagation characteristics in single core SF^ cable can now be modelled by the surge impedance of 61.4 ft,(typically about 60 to 75 ft), and wave propagation velocity, (typically 300 m/ys). A numerical simula tion of overvoltage wave-shape .in the receiving end of a SF^ cable joining to a overhead transmission line is simulated. The resulting voltage in the open-circuited SF^ cable receiving end rises in a staircase fashion of diminishing amplitude, to a value of 2 p.u. (See Figure 3.9). This can be explained by using the reflection (C.) and refraction coefficient (C ) of the system at the line-cable junction and is. 39 the open-circuited cable end respectively. For the line-cable junction at A, one has Z2 = 312, Z1 = 60ft Z - Z (Wave incident from cable) c^= z + 7^ = 312 '+" 60 = (3-7±) (Wave incident from line) Z2 = 60, Z = 312 ft 85 Figure'3.9: Overvoltage waveshapes at both ends of SF^ cable joining from overhead transmission line. 86 For the open end of the cable, we have 2 1 °° - 60 °° + 60 = 1 and C ,o R 2 x °° + 60 2 (Wave incident from cable, Z = 60, Z = ») Thus, the discrete rise in voltage wave shape can be expressed as v = 2 x .32 (1 + C.+ C.2 + . . .) where each step addition accurs at discrete time intervals of 2 travel times. On the other hand, this overall rise in overvoltages wave shape also agrees with the general exponential rise wave shape in charging of a capacitor. This is due to the inherent large self capacitance of cables. The overall rise in wave shape can : be sketched by modelling the SFg cable as a lumped capacitor equivalent to the total capacitance for the length of cable, and ignoring the surge impedance of the cable (See Figure 3.9). 87 CHAPTER 4: CORONA ATTENUATION AND DISTORTION CHARACTERISTICS OF LIGHTNING OVERVOLTAGE IN OVERHEAD TRANSMISSION LINES. 1. Introduction As the lightning voltage wave travels down the overhead transmission line, a high electric field is produced on the line conductor surface. When kV the electric field intensity exceeds the breakdown strength of air (^30 /em), ionization of surrounding air molecules takes place. This phenomenon will dissipate the unwanted surge energy away from the system and thus reduces the magnitude and initial rate of rise o? the lightning overvoltage. In transient lightning overvoltage studies, several numerical methods 44 have been employed to account for corona effects. Brown applied the concept of corona radius to account for the corona envelope produced on the conductor surface. The coronated line capacitances at higher voltages are also obtained '• o 12 by extrapolation. Darveniza also used lower wave propagation velocities higher voltages and different corona correction factor for different conduc tor configuration. However, both methods are not straightforward and are 43 not totally successful in duplicating field rest results. Umoto and Hara also transformed the transmission line equation for coronated lines into difference algebraic equations. However, this numerical approach is not efficient enough. Thus, an efficient and accurate numerical model for corona must be developed to predict the corona attenuation and distortion character istics on lightning overvoltage propagations in overhead lines. 2. Physical properties of corona attenuation and distortion characteristics The physical aspects and laws governing the behaviour of corona dis charge have been investigated since the beginning of this century. However, most of the investigations and applications have been limited to power frequency steady state or at most to switching transient conditions. From 88 the published field measurements for lightning surges, it can be observed that the attenuation resulting from corona effects is much larger than that resulting from transmission line series resistance losses. The non-linear characteristics of the corona discharge can be considered as (see Figure 4.1): a) Corona attenuation loss - From the quadratic law of corona loss pro-41 posed by Peek , the loss (v i^) per unit length is proportional to the square of the voltage above the critical corona voltage v i.e. where k = a • /-^r x 10 /m r , h = radius and height of conductor respectively a = Corona loss constant determined experimentally This corona attenuation loss can be modelled with a resistive current loss i^ through the corona resistive branch to ground as b) Increase in shunt capacitance - the retardation of the wave front by 42 corona can be explained by an increase in shunt capacitance. Skilling 43 and Umoto suggested that the increase in shunt capacitance is proportional to the voltage above the critical voltage Vcq, i.e. V C = 2k (1 - (4.3) corona c V where k = a x 10 h c c v 2h a = corona loss constant^determined experimentally c 89 V - VR - vc transmission line Corona shunt capacitance V / C = 2k (1 „ corona c V. 1 + K c R G =kn--~f corona R V rrfn rmr Figure 4.1: Nonlinear corona losses model. 90 This increase in capacitance can be modelled by : a capacitance branch to ground with the capactive current loss i„ Corona discharge only occur if the voltage is greater or equal to the critical corona voltage, and if the voltage increase with time, i.e. j 3v v £ v , and — > o. co 3t This is due to the fact that, when the voltage begins to decrease, the space charge consisting of heavy ions in the ionization region remains practically constant in magnitude and position during a short period of time. This slow diffusion of ions results in little energy loss in the case of decreasing voltage conditions even when v > VCQ. 3. Transmission line equations for coronated lines. The corona phenomena can now be described by the modified line equations. With the introduction of digital computers, these phenomena can be studied accurately by solving the equations describing the electromagnetic wave propagations taking corona into account as follows: = £ + (i-^S + v1-'^2-' <4-6> extra shunt Extra shunt capacitance conductance due to due to corona corona 43 45 Umoto and Inoue solved the above equations by the difference method. The line equations (4.5) and (4.6) are transformed into algebraic equations of small increments of distance, Ax, and time, At. However, this method is not efficient to implement into the digital computer as the method requires Ax to be as small as 7 mr when using At = .01 ys. 4. Solution of line equation by compensation method with trapezoidal rules The line equations (4.5) and (4.6) with corona losses can be solved by the compensation method. In this method, the line equations are first solved without the extra corona terms. The Bergeron's method using travelling wave technique together with modal ahalysis(See Chapter 2) is applied. Then, the corona losses can be treated as non-linear shunt branches connected to ground,. .< The trapezoidal rule can then applied to obtain the total current loss of the corona phenomena. By applying the trapezoidal rule of linear interpolation to the corona resistive branch to ground, we have j as shown in Figure 4.2, ±-e- vt+At= i W + (vt" i V where d = + — (4.7) v — v t + At t as d = slope of graph at time t. Also, d can be obtained by considering the equation (4.2) 92 Current i(v) ( Voltage v Figure 4.2: Linear interpolation for resistance corona branch. 8v/9t=v voltage v or current i Figure 4.3: Linear interpolation for capacitive corona branch. 93 ( v - v )' t CO t R v. or co k^ vt + k„ — - 2k„ v R v ~"R co di d = dv v 2 K. kR v 2 (4.8) Thus, we eventually have 1 . , ,v - 1 i v Vt.+ At " d \ + At + ( fc d ° (4.9) \ ' it + At + Vo (4.10) where v = v - v i , ( known from past history at time t) o t d t and ^R = a" (known from past history at time t) Similarly, since the corona capacitive branch current loss is given by 2k a c , , 9v x = (v - v ) — v co 3t or 9v v i 3t 2k (v - v ) c CO f(v, i) Applying linear interpolation of the 2 variables (i.e. from first term of Taylors' series), we have as shown in Figure 4.3 f(v, i) = f(v, i) t+At + 3v (v t+At \ , 3f ' Vt> + 31 ^t+At " ±t) (4.11) 94 3f 3v i (v v ) - v 2k t c CO -X, V CO 2kc <vt - vco)2 (4.12) and 3f 3i 2kj;v - v ) t * co 2H\ - \o> (4.13) Thus, we obtain f(v, i) t+At 3v 3t t+At v — v t+At t "At v. 2k(v - v ) v t+At /t ^ t co / Re-arranging equation (4.14) will give the linearized equation as Vt+At = Rc S+At + Vl (4.15) where R = 1 + v i * . . co t At 2k (v - v )2 C t CO * v At t (known from past 2k (v - v ) , . „ . c t co hxstory) and 1 + v K' A«-co t At 2k (v - v )2 c t co . .V XV At co t t ( Vt + 2k (v - v )2> from c t co past hxstory) Combining equations (4.10) and (4.15) for the voltages and currents in both corona resistive and capacitive branches by taking into account v = v_. = v c R and i = i + i„ c R we can obtain (^ + ^) v- (ic + iR) + (^ + ^) of or v = R'i + k' (4.17) Rc *R where R' and k* = Rc + Rk *R Vl + Rc Vo R Having the corona loss branches represented by a linear model as described in equation (4.17), the compensation method can then be applied to solve the transmission line equations including corona losses. In the compensation method, the transmission line is first reduced to a Thevinin equivalent (See Figure 4.4) and is described by vv= VQ + A2±, (4.18) where is a negative number. Then, this equation is solved simultaneously with the linearized equation for corona loss, as in equation (4.17). Thus, the resulting corona voltage and discharge current can be obtained as A m (m is ground) A : Thevinin equivalent network for transmission line without corona losses B : Nonlinear corona losses model v = R' i + k' Voltage v. v corona «• v =v +A i •»vkm o 2 Current i km i corona Figure 4.4: Compensation method for non-linear corona model 1corona - R' - A. (4.19) R'v - A-k* A o 2 and v corona • R1 - A. (4.20) 5. Influence on corona by adjacent sub-conductors in the same bundle Extra-high voltage phase conductors are designed to consist of several sub-conductors bundled together in order to reduce corona losses. The electric field on a sub-conductor surface is affected appreciably by the adjacent sub-conductors in the same bundles. The corona phenomenon is consequently influenced. The electric field on the sub-conductor surface due to the sub-46 conductor itself is given by max 2 IT e r o , Qr-= charge/length cv 2 IT e r o (4.21) where c = effective capacitance/length v = voltage of conductor r = radius of sub-conductor However, for a bundled conductor with 4 individual sub-conductors, the maximum electric field is given by^(See Figure 4.5) max Q1 ,- 1 2TT e + s72 + 2 -• sin 45u) -21 •- -2-TT e r o (4.22) 98 \ax - -2^tT ( 1 +vfs } C V „, effective where Q = ^ Figure 4.5: Critical voltage calculation by evaluation of maximum electric field on a 4-conductor bundle. After the maximum electric field on the conductor surface is obtained, the critical voltage for corona discharge can be computed by 6 equating the maximum electric field to 30 kV/cm or 3 x 10 v/m, . the electric breakdown strength in air. A typical critical voltage for a single conductor has been found to be 277 kV, and that for a 4-conductor bundle to be 558 kV. 6. Influence on corona by adjacent phase conductors Since the conductors in each phase are mutually coupled to one another, voltages are always induced in the adjacent conductors. Thus the maximum electric field on the conductor surface is affected. However, due to the design of transmission lines for extra high voltage levels, separating between phase conductors are usually large compared with radius of individual conductors. This effect usually change the overall critical overvoltages by less than 10%. But this change in critical voltage produces negligible effects on the overall corona attenuation and distortion characteristics on overvoltage wave (See Figure 4.6). 7. Optimal lumping locations and number of corona branch legs The equations with corona phenomenon is now solved by the dbmbensation method with the corona loss legs lumped at a few places along the transmission•-• line. However, the optimal locations and optimal number of lumped elements has to be determined. At first, 20 corona loss branches 70 m apart from one another were lumped between five transmission towers. Then, a separation of 350 m between the corona loss branches was used. This increase in separa tion increased the deviation of the predicted wave shape from field measure ments' appreciably (See Figure 4.7), from about 5 to 10%. This suggests that field measurements v ±t±cai = 303-kV (include effect of. cri xca adjacent phase conductors)) v . . 277 kV (neglect effect of crxtical adjacent phase conductors) Figure 4.6: Effect of adjacent phase conductors on corona losses. Figure 4.7: Effect of lumping distances on corona. 102 the optimal separation should be about 70 m. The field measurement for a 4-conductor bundled was then simulated. In using the corona loss constants (for case of 1-conductor bundle) a = 30 c a- = 10 x 106 slightly higher overvoltages were obtained. Then, a new set of corona constants (for case of 4-conductors bundle) a = 30 c a = 20 x 106 was used to give results consistent with those from field measurements (See Figure 4.8). Finally, the negative impulse overvoltage was also simulated for the 4-conductor bundle case. The corona loss in this case was found to be much less than the positive Impulse case. The corona loss constants were determined to be a = 15 c a = 10 x 106 With these sets of corona constants, the field test measurement was again replicated closely (See Figure 4.9). 8. Overall numerical modelling for corona effects The field test results of corona attenuation and distortion characteristics on a 500 kV test line were replicated by the method Overvoltages(kV) Figure 4.8: Positive impulse on 4- conductor bundle." Overvoltages(kV) 2000 FDUR-CONDUCTOR BUNDLE 1 2 3 4 5 6 field measurements a =15, a =10xl06 c G r a =15, a =5x10 c G Figure 4.9: Negative impulse on 4T conductor bundle. 105 developed earlier. This method examined corona characteristics in both single and bundled conductor lines. From the performed study, one can conclude that the effects of bundling of conductors is efficient in increas ing critical corona voltage. Furthermore, influence of adjacent phase conductors is negligible on corona effects. Thus, it is concluded that single phase line representation is sufficient for corona studies. Finally, it is determined that separation between the corona loss leg can be lumped at 70 m without sacrificing a loss of accuracy on the predicted coronated waveform. A. reduction in distance between corona legs will not improve the accuracy of the simulated results. It should be noted that lightning strokes will rarely hit more than one conductor at one time; thus corona phenomena have only been included for one conductor in this thesis, rather than for all three phases simultaneously. CHAPTER 5: CONCLUSIONS 106 The attentuation and distortion of lightning overvoltage waves on multi-phase transmission lines and multi-phase single core SF^ cables in compressed SF^ gas-insulated substations was studied. Corona effects of lightning overvoltages on overhead lines were also investigated. Available field test results for corona effects were duplicated to within 5% accuracy. Results obtained with the techniques developed by the author are 21 useful for lightning insulation co-ordination studies and other related ,. 13,14,40 ^ , i t J studies . xhe lightning surge wave front can be calculated at any location inside the substation, eg., inside the SF^ bus or at the trans-o former terminal. Based on the studies described in the thesis the following recommendations are made for future insulation co-ordination design studies: 1. Multi-phase untransposed lines can be represented by single-phase line models using self parameters calculated at a high frequency of approximately 1 M Hz (See Table 2.4). Series resistance should be ignored. Frequency dependent effects are not important for propa gation over distances less than 2 km. 2. Corona effects are important in reducing the magnitude and rate of rise of the incoming lightning overvoltage surge. Efficient solution techniques using compensation methods are developed to solve the non linear corona attenuation and distortion phenomenon. 3. Multi-phase SF^ single core cables can be represented by single phase cable models. Series resistance can be ignored. Cable parameter can be obtained with the simple formula for a go-return circuit for a coaxial cable with sufficient accuracy. 107 APPENDIX A: SKIN DEPTH ATTENUATION IN CONDUCTING MEDIUM WITH FINITE CONDUCTIVITY. This section shows that the core current return characteristics through the sheath for the SF^ cable could be obtained by a different approach. From the Maxwell's equations in a conducting medium, we have^ V x E = -jwu H (A.l) V x H = jtoeE + aE = aE, for good conductors (A.2) where a is conductivity of medium. From equations (A.l) and (A.2), we can get 2 2 VxVxE=V(V«E)-VE =-VE (for homogeneous medium) = -joiyV x H = -jwyaE = -m2E where m = /jtoya = ^ ^ • /u>ya This equation is identical to the diffusion equation with solutions — = E = E e-mZ, (A.3) a x o = E e-V"2~ Z • e-V~T Z (A.4) o = E e-jZ/6 • e"jZ/6 (A.5) o 108 where 6 = ~-. = -1 7= skin depth Thus, the tangential electric field E or the tangential current density Jx will be attenuated by ^ = -368 when the depth of penetration Z equals to the skin depth. For aluminum, we have the skin depth 6 as <5 = 7== (A.6) /irf (4ITX10-7) (3-8x10/) 8-1 ,— cm Therefore, for frequency above 1 kHz, the electric field is essentially attenuated and negligible flux outside the sheath. Thus, since character istic frequencies of lightning strokes exceeds 1 kHz, the above results indicate that each phase of the cable is decoupled from other phases as was shown previously in Chapter 3. After the tangential current density for one medium is obtained by equations (A.3) to (A.5), the tangential current density for another medium on the boundary to the first medium can be obtained by Elt = E2t °1 Jlt = ~2 J2t Thusm the total current flowing in different components of the cable system can be obtained by I = /JdA 109 BIBLIOGRAPHY 1. W. Diesendorf, 'Insulation coordination in high voltage electric power systems', Butterworth Cp. London 1974. 2. M. Uman, 'Understanding lightning', Bek Technical Publication Inc., Pennsylvania, 1971. 3. 'Transmission and Distribution Reference Book', Westinghouse Electric Corporation, Pennsylvania,1964. 4. F. Popolansky, 'Frequency distribution of amplitude of lightning currents', Report No. SC 33-71 (WG01) IWD, presented at CIGRE Meeting, Lugano, 1971. 5. L.V. Bewley,'Travelling waves on transmission systems', Dover Publi cation, N.Y. 1951. 6. 'The Principle and Practice of Insulation Co-ordination', Canadian Standard Assoc. C308, presented to the CEA Meeting, March 1979. 7. 'Transmission line reference book - 345 kV and above', Electric Power Research Institute, Palo Alto, 1979. 8. G.W. Brown, 'Lightning Performance I - shielding failure simplified', IEEE Trans. PAS-97, pp. 33-38, January/February 1978. 9. K.C. Lee, 'Propagation of the wave front on untransposed overhead and underground transmission lines', UBC M.A.Sc. Thesis, April 1977. 10. J.C. Cronin, R.G. Colclaser, R.F. Lehman, 'Transient lightning over voltage protection requirement for a 500 kV gas-insulated substation', IEEE Trans. PAS-97, pp. 68-78, January/February 1978. 11. Mitsubishi Electric Corporation,.'A conditon of insulation co-ordination study on Revelstoke Power Station 500 kV System', B.C. Hydro and Power Authority Report HAT-5159-B, HAT-5319, August 27, 1979. 12. Darveniza et al, 'Modelling for lightning performance calculations', IEEE Trans. PAS-98, pp. 1900-19 8, November/December 1979. 13. K.C. Lee, H.W. Dommel, 'Additions of untransposed - distributed parameter line models to the UBC Electromagnetic Transient Program', Progress Report to the System Engineering Division of B.C. Hydro and Power Authority, August 1979, (56 pages). 14. K.C. Lee, H.W. Dommel, 'Addition of modal analysis to the UBC Line Constants Program' ibid., January 1980, (39 pages). 15. UBC Computing Centre, 'UBC CEIGN' April 1972. 16. P.C. Magnusson, 'Travelling waves on multi-conductor open-wire line -a numerical survey of the effect of frequency dependence on modal decomposition*, IEEE Trans. PAS-92, pp. 999-1008, 1973. 110 17. W.S. Meyer, H.W. Dommel, 'Numerical modelling of frequency - dependent transmission line parameters in an electromagnetic transients program', IEEE Trans. PAS-93, pp. 1401-1409, 1974. 18. J. Marti, UBC Ph.D.:Thesis (to be completed). 19. A. Ametani, 'Surge propagation on a 500 kV untransposed line: Field Test Results', CIGRE WG 13-05, 13-74 (WG'05) 36 IWD, October 1974. 20. A. Ametani, 'Surge propagation on Japanese 500 kV untransposed transmission line', Proc IEE-121, pp. 136-138, 1974. 21. K.K. Nishikawara, N. Cuk, H.W. Dommel, K.C. Lee, T. Nitta, Y. Shibuya, T. Tanabe, M. Imataki, 'Development of insulation co-ordination requirement for high voltage SFg CGI substations of B.C. Hydro and Power Authority', presented to Canadian Electrical Association Spring Meeting, Montreal, March 1980, also abstracted in Modern Power and Engineering p. 31 May 1980, (26 pages). 22. L.M. ..Wedepohl, 'Radio interference fields in multi-conductor over head transmission line', Proc. IEE pp. 1875-1879, 1969. 23. R.D. Handel, 'Design of the MICA Hydro Electric Project', IEEE Trans. PAS-96, pp. 1535-1543, September/October 1977. 24. M. Mitani, 'Analysis of lightning surge phenomena and examination of insulation co-ordination in a 3-core type Gl cable', presented to the 1979 IEEE PES Summer Meeting A79 401-1. 25. G.W. Brown, R.G. Rocamora, 'Surge propagation in the 3-phase pipe type cable, 'IEEE Trans. PAS-95, pp. 89-95, January/February 1976. 26. E. Comellini et al, ' A computer program for determining electrical resistance and reactance of any transmission line', IEEE Trans. PAS-92, pp. 308-314, March-April 1973. 27. K.0. Abledu, 'Impedance calculation of cable using sub-division of the cable conductors', UBC M.A.Sc. Thesis, September 1979. 28. E.R. Sunde, 'Earth conduction effects in transmission systems', Dover Publication, N.Y. 1968. 29. F. Pollaczek, 'Sur le champ produit par un conducteur simple infinement long parcouru par un courant alternatif' , Revue geti. Elec. vol. 29 pp. 851-867, 1931. 30. L.M. Wedepohl, D.J. Wilcox, 'Transient analysis of underground power transmission systems', Proc. IEE vol. 120, pp. 252-260, 1973. 31. A. Ametani, 'A general formulation of impedance and admittance matrix of cables', IEEE Trans. PAS-99, pp. 902-910, May/June 1980. Ill 32. A. Ametani, 'Performance of a theoretical investigation;in multi phase power system cables', Bonneville Power Administration Contract No. 14.03 6049 N, July 1976. 33. G. Bianchi, G. Luoni, 'Induced currents and losses in single-core submarine cables', IEEE Trans. PAS-95, pp. 49-58, January/February 1976. 34. H.W. Dommel, 'Line Constants of overhead lines and underground cables', UBC EE 553 notes, May 1978. 35. UBC Computing Centre, 'UBC Function CBESK', December 1977. 36. M. Abramowitz, I.A. Stegun, 'Handbook of Mathematical Functions', National Bureau of Standards, Applied Mathematics Series 55, November 1970. 37. S.A. Schelkunoff, 'The electromagnetic theory of coaxial transmission lines and cylindrical shells', Bell System Technical Journal, vol. 13, pp. 532-579, 1934. 38. T. Ono and H. Matsubara, 'Surge voltages in HVDC submarine cables', Electrical Engineering in Japan, vol. 94, pp. 42-49, 1974. 39. A. Greenwood, 'Electrical transients in Power systems', Wiley Interscience, N.Y. 1971. 40. K.C. Lee, H.W. Dommel, 'Calculation of cable parameters of a single core SF6 cable', Progress Report to the System Engineering Division of B.C. Hydro and Power Authority, April 1980. 41. F.W. Peek, 'Laws of corona I II III', - AIEE Trans, vol. 30, p. 1889, 1912. 42. H.H. Skilling et al, 'Distortion of travelling waves by corona', AIEE Trans, vol. 50, p. 850, 1937. 43. J. Umoto et al, 'Numerical analysis of surge propagation in single conductor system considering corona losses', Electrical Engineering in Japan, vol. 89, p. 21, 1968. 44. G.W. Brown, 'Lightning Performance II - updating backflash calculations', IEEE Trans. PAS-97, pp. 34-51, January/February 1978. 45. A. Inoue, 'High voltage travelling waves with corona discharge on bundle conductors', presented to the IEEE 1978 Winter Meeting A78 170-3. 46. W. Hayt, 'Engineering Electromagnetics', McGraw ,Hill , N.Y. 1*974. 47. H.W.Dommel, W.S.Meyer, 'Computation of electromagnetic transients' , Proc. IEEE vol.62, pp.983-993, July 1974. 48. H.W.Dommel, private communication July 1979. 49. G. Strang, 'Linear algebra and Its applications',Academic Press N.Y. 1976. 112 50. ,'A.T.Adams, ' Electromagnetics for engineers', Ronald Press Company, New York, N.Y. 1971. 51." G.L. Ford, E.P.Dick, ' Study of fault-produced oyervoltages as-related to grounding practise for SF -insulated substations', Canadian Electrical Association Contract No. 071T101, Sept. 1979. 52. ' Lightning protection design guide book for power stations and substations', Central Research Institute of Electric Power Industry CRIEPI Report No. 175034, Tokyo, Japan 1976. (in Japanese) 


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